Theorem zeqmsubd | index | src |

theorem zeqmsubd (G: wff) (a b c d n: nat):
  $ G -> modZ(n): a = b $ >
  $ G -> modZ(n): c = d $ >
  $ G -> modZ(n): a -Z c = b -Z d $;
StepHypRefExpression
1 zeqmtr
modZ(n): a -Z c = b -Z c -> modZ(n): b -Z c = b -Z d -> modZ(n): a -Z c = b -Z d
2 zeqmsub1
modZ(n): a -Z c = b -Z c <-> modZ(n): a = b
3 hyp h1
G -> modZ(n): a = b
4 2, 3 sylibr
G -> modZ(n): a -Z c = b -Z c
5 zeqmsub2
modZ(n): b -Z c = b -Z d <-> modZ(n): c = d
6 hyp h2
G -> modZ(n): c = d
7 5, 6 sylibr
G -> modZ(n): b -Z c = b -Z d
8 1, 4, 7 sylc
G -> modZ(n): a -Z c = b -Z d

Axiom use

axs_prop_calc (ax_1, ax_2, ax_3, ax_mp, itru), axs_pred_calc (ax_gen, ax_4, ax_5, ax_6, ax_7, ax_10, ax_11, ax_12), axs_set (elab, ax_8), axs_the (theid, the0), axs_peano (peano1, peano2, peano5, addeq, muleq, add0, addS, mul0, mulS)